On the stability of a strongly stabilizing control for degenerate systems in Hilbert spaces

• Autorzy: Hariri Mohamed , Bouteffal Zohra , Beghersa Nor-el-houda , Benabdallah Mehdi

Identyfikatory

DOI

10.1515/dema-2022-0238

• Języki publikacji: EN

Abstrakty

EN

In this article, we explain how a recent Lyapunov theorem on stability plays a role in the study of the strong stabilizability problem in Hilbert spaces. We explore a degenerate controlled system and investigate the properties of a feedback control to stabilize such system in depth. The spectral theory of an appropriate pencil operator is used to generate robustness constraints for a stabilizing control.

Słowa kluczowe

EN

stabilization of systems by feedback; operator theory; degenerate systems

PL

stabilność układów; sprzężenie zwrotne; teoria operatorów; systemy zdegenerowane

• Wydawca: De Gruyter
• Czasopismo: Demonstratio Mathematica
• Rocznik: 2023
• Tom: Vol. 56, nr 1
• Strony: art. no. 20220238
• Opis fizyczny: Bibliogr. 14 poz.

Twórcy

autor

Hariri Mohamed

• Department of Mathematics and Informatics, Faculty of Science and Technology, Belhadj Bouchaib University, Ain Temouchent, 46000, Algeria

autor

Bouteffal Zohra

• Department of Mathematics, Mustapha Stambouli University, Mascara, 29000, Algeria

autor

Beghersa Nor-el-houda

• Department of Mathematics, Faculty of Mathematics and Computer Sciences, USTOran, 31000, Algeria

autor

Benabdallah Mehdi

• Department of Mathematics, Faculty of Mathematics and Computer Sciences, USTOran, 31000, Algeria

Bibliografia

• [1] J. L. Daletckii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, American Math Society Providence, Rhode Island, 1975.
• [2] T. Roberto, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), no. 3, 383–403, DOI: https://doi.org/10.1016/0022-247X(75)90067-0.
• [3] A. V. Balakrishnan, Strong stabilizability and the steady state Riccati equation, Appl. Math. Optim. 7 (1981), no. 7, 335–345, DOI: https://doi.org/10.1007/BF01442125.
• [4] Z. Dastgeer, A. Youns, and C. Tunç, Distinguishability of the systems with regular pencil, Linear Algebra App. 652 (2022), no. 1, 82–96, DOI: https://doi.org/10.1016/j.laa.2022.07.004.
• [5] C. Tunç and O. Tunç, On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM. 115 (2021), no. 115, 1–17, DOI: https://doi.org/10.1007/s13398-021-01058-8.
• [6] C. Tunç, O. Tunç, Y. Wang, and J. Yao, Qualitative analyses of differential systems with time-varying delays via Lyapunov-Krasovski-approach, Mathematics 9 (2021), no. 1196, 1–20, DOI: https://doi.org/10.3390/math9111196.
• [7] V. I. Korobov and G. M. Sklyar, Strong stabilizability of contractive systems in Hilbert spaces, Differentsialnye Uranvneniya 20 (1984), no. 11, 1320–1326.
• [8] Y. K. Chang, A. Pereira, and R. Ponce, Approximate controllability for fractional differential equations of Sobolev type via properties on resolvent operators, Fract. Calc. Appl. Anal. 20 (2017), no. 4, 963–987, DOI: https://doi.org/10.1515/fca-2017-0050.
• [9] C. Dineshkumar, N. K. Sooppy, R. Udhayakumar, and V. Vijayakumar, A discussion on approximate controllability of Sobolev-type Hilfer neutral fractional stochastic differential inclusions, Asian J. Control 24 (2021), no. 5, 1–17, DOI: https://doi.org/10.1002/asjc.2650.
• [10] S. L. Gefter and A. L. Piven, Implicit linear non-homogeneous difference equation in Banach and locally convex spaces, Geometry 15 (2019), no. 3, 336–353, DOI: https://doi.org/10.15407/mag15.03.336.
• [11] F. R. Gantmakher, Theory of Matrices, Nauka, Moscow, 1988.
• [12] M. Benabdallah and M. Hariri, On the stability of the quasi-linear implicit equations in Hilbert spaces, Khayyam J. Math. 5 (2019), no. 1, 105–112, DOI: https://doi.org/10.22034/kjm.2019.81222.
• [13] M. Manafred and P. Vyacheslav, Spectral Theory of Operator Pencil, Hermite-Biehler Functions, and Their Applications, Springer International Publishing, Switzerland, 2015.
• [14] P. L. Butzer and H. Berens, Semigroups of Operators and Approximations, Springer-Verlag, New York, 1967.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr POPUL/SP/0154/2024/02 w ramach programu "Społeczna odpowiedzialność nauki II" - moduł: Popularyzacja nauki (2025).